Science The Hofstadter Butterfly

Published on May 15th, 2013 | by Michael Ricciardi


Long-Predicted Fractal Energy Pattern Observed For First Time By Physicists

The Hofstadter Butterfly

Hofstadter’s butterfly in color. The horizontal axis is the energy (or chemical potential) and the vertical axis is the magnetic flux through the unit cell. The warm and cold colors represent positive and negative values of Hall conductance, respectively (image credit: Mytomi)

Forty years ago, a mysterious and beautiful butterfly-shaped magnetic energy pattern was theorized by famed physicist and Pulitzer Prize winning writer Douglas Hofstadter. But due to previous limitations in the laboratory technology used to generate strong magnetic fields, this pattern had here-to-fore never materialized.

Today, two separate but cross-collaborating research teams — one of which is from the High Magnetic Field Laboratory (MagLab) at Florida State University (Tallahassee, Florida) — reported the first ever observations of the ‘Hofstadter butterfly’.

“The observation of the Hofstadter butterfly marks a real landmark in condensed matter physics and high magnetic field research,” said Greg Boebinger, director of the MagLab. “It opens a new experimental direction in materials research.”

Condensed matter physics is a branch of physics that seeks to understand the physical properties of condensed phases of matter. The most common condensed phases of matter are solids and liquids but modern research here mostly focuses on more “exotic” phases such as the low-temperature superconducting, ferromagnetic and anti-ferromagnetic phases of atomic spins (in crystal lattices) and the Bose-Einstein condensate found in cold atomic systems.

Fellow MagLab physicist Nicholas Bonesteel, added:

“The Hofstadter butterfly is a beautiful fractal energy pattern that has intrigued physicists for decades. Seeing clear experimental evidence for it is a real breakthrough.”

About the Butterfly

While a student at the University of Oregon, Hofstadter wrote what would become a very influential paper (“Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields”) in condensed matter physics. In the paper, he theorized and predicted certain allowable energy levels of a electron in a 2D square lattice (recently constructed from the above-mentioned materials) as a function of a magnetic field applied to the lattice, and forming what would later be known as a fractal set.

Rendering of the butterfly by Hofstadter

Rendering of the butterfly by Hofstadter

In this case, the fractal set describes the observed distribution of energy levels for small-scale changes in the (applied) magnetic field that recursively repeat patterns observed in the larger scale structure (note: this is known in fractal geometry as self-similarity). The graphical representation of this fractal structure or pattern is what came to be known as the  “Hofstadter’s butterfly”. It is one of only two examples of fractal geometry in physics (the other being the  KAM tori).

How The Break Through Was Achieved

This particular breakthrough required measurements of material samples at very low temperatures and subjected to very high magnetic field strengths (up to 35 Teslas; note: the most powerful MRI machines generate fields less than 2.0 Teslas). The MagLab is one of the few such research labs in the world capable of creating both conditions simultaneously.

The key to observing the unique butterfly energy pattern was the fabricating of a unique material composed of boron nitride (BN) and graphene — one atom thick sheets of carbon atoms that have become the wonder-nano material of the 21st Century. The stuff is in batteries, cell phones, televisions, computers…it is nearly transparent (being one atom layer thick) but is 300 times stronger than steel (for its weight) and 1000 times more conductive that silicon.

The material was first cooled and then exposed to the high magnetic field — producing the ‘Hofstadter butterfly’ energy pattern.

MagLab deputy director Eric Palm also commented on the discovery:

“This is about a puzzle that has been solved. It is really about scientific curiosity. It is an exciting confirmation of a theory that was made years ago.”

The Research Teams

One research team was led by Columbia University’s Philip Kim and included researchers from City University of New York, the University of Central Florida, Tohoku University and the National Institute for Materials Science in Japan. The team’s work will be published today in the Advanced Online Publication of the journal Nature.

Similar results were discovered at the MagLab (reported here) by a team led by Pablo Jarillo-Herrero and Raymond Ashoori at MIT, as well as scientists from Tohoku University and the National Institute for Materials Science in Japan. Their work (‘Massive Dirac Fermions and Hofstadter Butterfly in a van der Waals Heterostructure’) is also expected to be published soon.




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About the Author

Michael Ricciardi is a well-published writer of science/nature/technology articles as well as essays, poetry and short fiction. Michael has interviewed dozen of scientists from many scientific fields, including Brain Greene, Paul Steinhardt, Arthur Shapiro, and Nobel Laureate Ilya Progogine (deceased). Michael was trained as a naturalist and taught ecology and natural science on Cape Cod, Mass. from 1986-1991. His first arts grant was for production of the environmental (video) documentary 'The Jones River - A Natural History', 1987-88 (Kingston, Mass.). Michael is an award winning, internationally screened video artist. Two of his more recent short videos; 'A Time of Water Bountiful' and 'My Name is HAM' (an "imagined memoir" about the first chimp in space), and several other short videos, can be viewed on his website ( He is also the author of the (Kindle) ebook: Artful Survival ~ Creative Options for Chaotic Times

  • Narotam Lathia

    Although later experiments have revealed complex interactions, this
    state of matter was first predicted, generally, in papers by Satyendra Nath Bose and Albert Einstein in 1924–25. Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it. (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.[2]). Einstein then extended Bose’s ideas to material particles (or matter) in two other papers [3]. The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4 (4He),
    are allowed to share quantum states with each other. Einstein
    demonstrated that cooling bosonic atoms to a very low temperature would
    cause them to fall (or “condense”) into the lowest accessible quantum state, resulting in a new form of matter.

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